Abstract

We study the approximation theory of a special kind of neighborhood systems, called total pure reflexive neighborhood systems, which is a generalization of pretopological and topological neighborhood systems. In the framework of such neighborhood systems, the so-called lower and upper approximations are considered. For a pretopological neighborhood system on a fixed nonempty set $U$, we show that the family ${\mathcal T}$ of fixed points of its lower approximation is a topology for $U$, and establish a characterization (in terms of neighborhoods) of ${\mathcal T}$-open sets.

We then regard a reflexive relation $R$ on $U$ as the total pure reflexive neighborhood system $x \longmapsto\{ R(x)=\{ \ y \in U \mid (x,y) \in R \}\}$, whose induced upper approximation is identical to the commonly used upper approximation ${R^*}: 2^U \to2^U$ based on $R$. We show that the family ${\mathcal T}_R$ of all subsets $X$ of $U$ for which ${R^*}(U-X)=U-X$ is an Alexandroff topology for $U$, and that the pre-topologically maximal neighborhood system of the neighborhood system $x \longmapsto\{ R(x) \}$ associated to $R$ is exactly the pretopological neighborhood system whose induced upper approximation is identical to ${R^*}: 2^U \to2^U$. Accordingly, we show that for each $x \in U$, its smallest ${\mathcal T}_R$-open neighborhood is the intersection of all ${\mathcal T}_R$-open sets containing $R(x)$. In addition, we study the so-called $R$-definability. We establish a characterization of $R$-definable sets in terms of ${R^*}$ and its dual ${R_*}$, and present a necessary condition for $R$-definability.